The Banach-Mazur Game
Around 1928, the Polish mathematician S. Mazur invented the following mathematical “game.” Player (A) is “dealt” an arbitrary subset A of a closed interval I0. The complementary set B = I0 − A is dealt to player (B). The game 〈A, B〉 is played as follows: (A) chooses arbitrarily a closed interval I1⊂I0; then (B) chooses a closed interval I2⊂I1; then (A) chooses a closed interval I3⊂I2; and so on, alternately. Together the players determine a nested sequence of closed intervals In, (A) choosing those with odd index, (B) those with even index. If the set ∩I n has at least one point in common with A, then (A) wins; otherwise, (B) wins.
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